Since the 1990s, chaos has become one of the most active research areas in nonlinear science. In addition to the extensive study on the complex and rich dynamics of chaos, its potential applications have also been widely explored. It is particularly noticed that, due to the peculiar properties of chaotic systems, a lot of synchronization-based cryptographic designs have been proposed. As a common challenge for all cryptographic designs, the chaos-based cryptosystems attract different kinds of attacks at nearly the same time of their launches. However, due to the nature of chaos, the attacks are specifically designed for a particular cryptosystem, so that the weaknesses of the systems can be revealed and identified. In the present research study, it is to develop a systemic design of adaptive observers for nonlinear systems, aiming at testifying the security of chaos-synchronization-based cryptosystems in terms of their anti-adaptiveness. It is served as a general attack, which, in turn, becomes a pre-design criterion for a chaos-synchronization-based cryptosystem. Designing adaptive observers for nonlinear systems is not new in the domain of modern control theory. However, due to the nonlinearity and complexity of the chaotic systems, existing approaches can only address some restricted system forms or configurations, while most of the technical challenges remain unresolved today. It is particularly true for the case of the proposed adaptive attack, where a very limited number of system states are observable, while multiple parameters are unknown in the system. The proposed design is based on a combination of linear feedback control and dynamic minimization. Its global convergence is proved by the Lyapunov stability theorem when the unknown parameters reside in the dynamic equations of measurable states only. For a more general case, when unknown parameters appear in the unobservable system state, the convergence is justified by some local approaches. The design is applied for system identifications of some practical systems to illustrate its effectiveness. Being the main concern of this thesis, different kinds of chaos-synchronization-based communication schemes are studied. By using the proposed adaptive observers as a systematic attack, it is demonstrated that the states and/or system parameters, as well as the transmitting messages, can be revealed from the transmitted signals. It hence implies that the securities of many proposed chaos-synchronization-based cryptosystems are in fact questionable, and the design proposed here can act as a test for a cryptographic design so that a sufficient security level can be ensured.